given that S is the upper half surface of the sphere of radius one centered at the
origin and C is its boundary.
(1) Find the eigenvalues and the corresponding eigenvectors of A.
(2) Find an orthogonal matrix X that diagonalizes A to a diagonal matrix D .
(3) Express A−1 in terms of X and D .
(4) Find the determinant of A4 .
Consider a linear-regression problem y=ax+b, where the slope a and the
intercept b are parameters to estimated. Measurements were performed at
x = −1, 0, 1, 2 and the resulting y coordinates are y = 0.7, 1.6, 2.2, 3.0
respectively. Hence this problem can be modeled as a linear system:
(1) Construct the system matrix A.
(2) How would you describe this system? (multiple choices)
(A) Underdetermined; (B) Determined;
(C) Overdetermined; (D) Consistent;
(E) Inconsistent.
A string is clamped at both ends x = 0 and x = L . We assume that the vibration is of
small amplitude and satisfies the wave equation,
where v is the wave velocity. The string is set in vibration with the following initial
conditions:
where v0 = constant . Solve for y(x,t) . [Hint: let y(x,t) = X(x)T(t)]
(3) Three radioactive nuclei decay successively in series such that the number Ni(t)
of three types obey the equations:
If initially N1 = N , N2 = 0 , and N3 = n , find N3 (t) by using the Laplace
transform.
(2) Use the method of Fourier transform to solve the above differential equation.
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